3.2306 \(\int \frac{(1+2 x)^{5/2}}{\left (2+3 x+5 x^2\right )^2} \, dx\)

Optimal. Leaf size=283 \[ -\frac{(5-4 x) (2 x+1)^{3/2}}{31 \left (5 x^2+3 x+2\right )}-\frac{8}{155} \sqrt{2 x+1}+\frac{1}{155} \sqrt{\frac{1}{310} \left (10325 \sqrt{35}-32678\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )-\frac{1}{155} \sqrt{\frac{1}{310} \left (10325 \sqrt{35}-32678\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )-\frac{1}{155} \sqrt{\frac{2}{155} \left (32678+10325 \sqrt{35}\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )+\frac{1}{155} \sqrt{\frac{2}{155} \left (32678+10325 \sqrt{35}\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right ) \]

[Out]

(-8*Sqrt[1 + 2*x])/155 - ((5 - 4*x)*(1 + 2*x)^(3/2))/(31*(2 + 3*x + 5*x^2)) - (S
qrt[(2*(32678 + 10325*Sqrt[35]))/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[
1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/155 + (Sqrt[(2*(32678 + 10325*Sqrt[35]))/15
5]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]]
)/155 + (Sqrt[(-32678 + 10325*Sqrt[35])/310]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35
])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/155 - (Sqrt[(-32678 + 10325*Sqrt[35])/310]*Log
[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/155

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Rubi [A]  time = 1.29951, antiderivative size = 283, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ -\frac{(5-4 x) (2 x+1)^{3/2}}{31 \left (5 x^2+3 x+2\right )}-\frac{8}{155} \sqrt{2 x+1}+\frac{1}{155} \sqrt{\frac{1}{310} \left (10325 \sqrt{35}-32678\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )-\frac{1}{155} \sqrt{\frac{1}{310} \left (10325 \sqrt{35}-32678\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )-\frac{1}{155} \sqrt{\frac{2}{155} \left (32678+10325 \sqrt{35}\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )+\frac{1}{155} \sqrt{\frac{2}{155} \left (32678+10325 \sqrt{35}\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 + 2*x)^(5/2)/(2 + 3*x + 5*x^2)^2,x]

[Out]

(-8*Sqrt[1 + 2*x])/155 - ((5 - 4*x)*(1 + 2*x)^(3/2))/(31*(2 + 3*x + 5*x^2)) - (S
qrt[(2*(32678 + 10325*Sqrt[35]))/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[
1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/155 + (Sqrt[(2*(32678 + 10325*Sqrt[35]))/15
5]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]]
)/155 + (Sqrt[(-32678 + 10325*Sqrt[35])/310]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35
])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/155 - (Sqrt[(-32678 + 10325*Sqrt[35])/310]*Log
[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/155

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Rubi in Sympy [A]  time = 77.8175, size = 391, normalized size = 1.38 \[ - \frac{\left (- 4 x + 5\right ) \left (2 x + 1\right )^{\frac{3}{2}}}{31 \left (5 x^{2} + 3 x + 2\right )} - \frac{8 \sqrt{2 x + 1}}{155} - \frac{\sqrt{14} \left (- \frac{97 \sqrt{35}}{5} + 14\right ) \log{\left (2 x - \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{2170 \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{14} \left (- \frac{97 \sqrt{35}}{5} + 14\right ) \log{\left (2 x + \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{2170 \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{35} \left (\frac{28 \sqrt{10} \sqrt{2 + \sqrt{35}}}{5} - \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (- \frac{194 \sqrt{35}}{5} + 28\right )}{10}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} - \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{1085 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{35} \left (\frac{28 \sqrt{10} \sqrt{2 + \sqrt{35}}}{5} - \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (- \frac{194 \sqrt{35}}{5} + 28\right )}{10}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} + \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{1085 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+2*x)**(5/2)/(5*x**2+3*x+2)**2,x)

[Out]

-(-4*x + 5)*(2*x + 1)**(3/2)/(31*(5*x**2 + 3*x + 2)) - 8*sqrt(2*x + 1)/155 - sqr
t(14)*(-97*sqrt(35)/5 + 14)*log(2*x - sqrt(10)*sqrt(2 + sqrt(35))*sqrt(2*x + 1)/
5 + 1 + sqrt(35)/5)/(2170*sqrt(2 + sqrt(35))) + sqrt(14)*(-97*sqrt(35)/5 + 14)*l
og(2*x + sqrt(10)*sqrt(2 + sqrt(35))*sqrt(2*x + 1)/5 + 1 + sqrt(35)/5)/(2170*sqr
t(2 + sqrt(35))) + sqrt(35)*(28*sqrt(10)*sqrt(2 + sqrt(35))/5 - sqrt(10)*sqrt(2
+ sqrt(35))*(-194*sqrt(35)/5 + 28)/10)*atan(sqrt(10)*(sqrt(2*x + 1) - sqrt(20 +
10*sqrt(35))/10)/sqrt(-2 + sqrt(35)))/(1085*sqrt(-2 + sqrt(35))*sqrt(2 + sqrt(35
))) + sqrt(35)*(28*sqrt(10)*sqrt(2 + sqrt(35))/5 - sqrt(10)*sqrt(2 + sqrt(35))*(
-194*sqrt(35)/5 + 28)/10)*atan(sqrt(10)*(sqrt(2*x + 1) + sqrt(20 + 10*sqrt(35))/
10)/sqrt(-2 + sqrt(35)))/(1085*sqrt(-2 + sqrt(35))*sqrt(2 + sqrt(35)))

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Mathematica [C]  time = 0.516641, size = 147, normalized size = 0.52 \[ -\frac{\sqrt{2 x+1} (54 x+41)}{155 \left (5 x^2+3 x+2\right )}+\frac{2 \left (97 \sqrt{31}-264 i\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2-i \sqrt{31}}}\right )}{155 \sqrt{-155 i \left (\sqrt{31}-2 i\right )}}+\frac{2 \left (97 \sqrt{31}+264 i\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2+i \sqrt{31}}}\right )}{155 \sqrt{155 i \left (\sqrt{31}+2 i\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + 2*x)^(5/2)/(2 + 3*x + 5*x^2)^2,x]

[Out]

-(Sqrt[1 + 2*x]*(41 + 54*x))/(155*(2 + 3*x + 5*x^2)) + (2*(-264*I + 97*Sqrt[31])
*ArcTan[Sqrt[5 + 10*x]/Sqrt[-2 - I*Sqrt[31]]])/(155*Sqrt[(-155*I)*(-2*I + Sqrt[3
1])]) + (2*(264*I + 97*Sqrt[31])*ArcTan[Sqrt[5 + 10*x]/Sqrt[-2 + I*Sqrt[31]]])/(
155*Sqrt[(155*I)*(2*I + Sqrt[31])])

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Maple [B]  time = 0.036, size = 485, normalized size = 1.7 \[ 16\,{\frac{1}{ \left ( 1+2\,x \right ) ^{2}-8/5\,x+3/5} \left ( -{\frac{27\, \left ( 1+2\,x \right ) ^{3/2}}{3100}}-{\frac{7\,\sqrt{1+2\,x}}{1550}} \right ) }+{\frac{101\,\sqrt{20+10\,\sqrt{35}}\sqrt{35}}{48050}\ln \left ( 5+10\,x+\sqrt{35}-\sqrt{1+2\,x}\sqrt{20+10\,\sqrt{35}} \right ) }-{\frac{132\,\sqrt{20+10\,\sqrt{35}}}{24025}\ln \left ( 5+10\,x+\sqrt{35}-\sqrt{1+2\,x}\sqrt{20+10\,\sqrt{35}} \right ) }+{\frac{ \left ( 2020+1010\,\sqrt{35} \right ) \sqrt{35}}{24025\,\sqrt{-20+10\,\sqrt{35}}}\arctan \left ({\frac{1}{\sqrt{-20+10\,\sqrt{35}}} \left ( 10\,\sqrt{1+2\,x}-\sqrt{20+10\,\sqrt{35}} \right ) } \right ) }-{\frac{5280+2640\,\sqrt{35}}{24025\,\sqrt{-20+10\,\sqrt{35}}}\arctan \left ({\frac{1}{\sqrt{-20+10\,\sqrt{35}}} \left ( 10\,\sqrt{1+2\,x}-\sqrt{20+10\,\sqrt{35}} \right ) } \right ) }+{\frac{8\,\sqrt{35}}{155\,\sqrt{-20+10\,\sqrt{35}}}\arctan \left ({\frac{1}{\sqrt{-20+10\,\sqrt{35}}} \left ( 10\,\sqrt{1+2\,x}-\sqrt{20+10\,\sqrt{35}} \right ) } \right ) }-{\frac{101\,\sqrt{20+10\,\sqrt{35}}\sqrt{35}}{48050}\ln \left ( 5+10\,x+\sqrt{35}+\sqrt{1+2\,x}\sqrt{20+10\,\sqrt{35}} \right ) }+{\frac{132\,\sqrt{20+10\,\sqrt{35}}}{24025}\ln \left ( 5+10\,x+\sqrt{35}+\sqrt{1+2\,x}\sqrt{20+10\,\sqrt{35}} \right ) }+{\frac{ \left ( 2020+1010\,\sqrt{35} \right ) \sqrt{35}}{24025\,\sqrt{-20+10\,\sqrt{35}}}\arctan \left ({\frac{1}{\sqrt{-20+10\,\sqrt{35}}} \left ( 10\,\sqrt{1+2\,x}+\sqrt{20+10\,\sqrt{35}} \right ) } \right ) }-{\frac{5280+2640\,\sqrt{35}}{24025\,\sqrt{-20+10\,\sqrt{35}}}\arctan \left ({\frac{1}{\sqrt{-20+10\,\sqrt{35}}} \left ( 10\,\sqrt{1+2\,x}+\sqrt{20+10\,\sqrt{35}} \right ) } \right ) }+{\frac{8\,\sqrt{35}}{155\,\sqrt{-20+10\,\sqrt{35}}}\arctan \left ({\frac{1}{\sqrt{-20+10\,\sqrt{35}}} \left ( 10\,\sqrt{1+2\,x}+\sqrt{20+10\,\sqrt{35}} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+2*x)^(5/2)/(5*x^2+3*x+2)^2,x)

[Out]

16*(-27/3100*(1+2*x)^(3/2)-7/1550*(1+2*x)^(1/2))/((1+2*x)^2-8/5*x+3/5)+101/48050
*ln(5+10*x+35^(1/2)-(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(20+10*35^(1/2))^(1/2)
*35^(1/2)-132/24025*ln(5+10*x+35^(1/2)-(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(20
+10*35^(1/2))^(1/2)+101/24025/(-20+10*35^(1/2))^(1/2)*arctan((10*(1+2*x)^(1/2)-(
20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(20+10*35^(1/2))*35^(1/2)-264/24
025/(-20+10*35^(1/2))^(1/2)*arctan((10*(1+2*x)^(1/2)-(20+10*35^(1/2))^(1/2))/(-2
0+10*35^(1/2))^(1/2))*(20+10*35^(1/2))+8/155/(-20+10*35^(1/2))^(1/2)*arctan((10*
(1+2*x)^(1/2)-(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*35^(1/2)-101/4805
0*ln(5+10*x+35^(1/2)+(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(20+10*35^(1/2))^(1/2
)*35^(1/2)+132/24025*ln(5+10*x+35^(1/2)+(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(2
0+10*35^(1/2))^(1/2)+101/24025/(-20+10*35^(1/2))^(1/2)*arctan((10*(1+2*x)^(1/2)+
(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(20+10*35^(1/2))*35^(1/2)-264/2
4025/(-20+10*35^(1/2))^(1/2)*arctan((10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-
20+10*35^(1/2))^(1/2))*(20+10*35^(1/2))+8/155/(-20+10*35^(1/2))^(1/2)*arctan((10
*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*35^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x + 1)^(5/2)/(5*x^2 + 3*x + 2)^2,x, algorithm="maxima")

[Out]

integrate((2*x + 1)^(5/2)/(5*x^2 + 3*x + 2)^2, x)

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Fricas [A]  time = 0.278359, size = 1297, normalized size = 4.58 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x + 1)^(5/2)/(5*x^2 + 3*x + 2)^2,x, algorithm="fricas")

[Out]

-1/19070708650*4805^(3/4)*sqrt(826)*sqrt(31)*(4805^(1/4)*sqrt(826)*sqrt(31)*(516
25*sqrt(7)*(54*x + 41) - 32678*sqrt(5)*(54*x + 41))*sqrt(2*x + 1)*sqrt((32678*sq
rt(7)*sqrt(5) - 361375)/(674800700*sqrt(7)*sqrt(5) - 4799048559)) + 1149356*1193
983^(1/4)*sqrt(5)*(5*x^2 + 3*x + 2)*arctan(64015*1193983^(1/4)*sqrt(31)*(505*sqr
t(7) - 264*sqrt(5))/(4805^(1/4)*sqrt(1829)*sqrt(826)*sqrt(31)*(51625*sqrt(7) - 3
2678*sqrt(5))*sqrt(sqrt(5)*(1193983^(1/4)*4805^(1/4)*sqrt(826)*(2857432837452652
734362918220094532883072047512224503*sqrt(7)*sqrt(5) - 1690462832043042559225674
7921672766019291595984758430)*sqrt(2*x + 1)*sqrt((32678*sqrt(7)*sqrt(5) - 361375
)/(674800700*sqrt(7)*sqrt(5) - 4799048559)) + 14455*sqrt(5)*(4437922563568471220
638688941619965174410065763500*sqrt(7)*sqrt(5)*(2*x + 1) - 525108919175847370910
97113282085264347697791933598*x - 2625544595879236854554855664104263217384889596
6799) + 14455*sqrt(7)*(4437922563568471220638688941619965174410065763500*sqrt(7)
*sqrt(5) - 26255445958792368545548556641042632173848895966799))/(443792256356847
1220638688941619965174410065763500*sqrt(7)*sqrt(5) - 262554459587923685455485566
41042632173848895966799))*sqrt((32678*sqrt(7)*sqrt(5) - 361375)/(674800700*sqrt(
7)*sqrt(5) - 4799048559)) + 64015*4805^(1/4)*sqrt(826)*sqrt(2*x + 1)*(51625*sqrt
(7) - 32678*sqrt(5))*sqrt((32678*sqrt(7)*sqrt(5) - 361375)/(674800700*sqrt(7)*sq
rt(5) - 4799048559)) + 1984465*1193983^(1/4)*(10*sqrt(7) - 97*sqrt(5)))) + 11493
56*1193983^(1/4)*sqrt(5)*(5*x^2 + 3*x + 2)*arctan(64015*1193983^(1/4)*sqrt(31)*(
505*sqrt(7) - 264*sqrt(5))/(4805^(1/4)*sqrt(1829)*sqrt(826)*sqrt(31)*(51625*sqrt
(7) - 32678*sqrt(5))*sqrt(-sqrt(5)*(1193983^(1/4)*4805^(1/4)*sqrt(826)*(28574328
37452652734362918220094532883072047512224503*sqrt(7)*sqrt(5) - 16904628320430425
592256747921672766019291595984758430)*sqrt(2*x + 1)*sqrt((32678*sqrt(7)*sqrt(5)
- 361375)/(674800700*sqrt(7)*sqrt(5) - 4799048559)) - 14455*sqrt(5)*(44379225635
68471220638688941619965174410065763500*sqrt(7)*sqrt(5)*(2*x + 1) - 5251089191758
4737091097113282085264347697791933598*x - 26255445958792368545548556641042632173
848895966799) - 14455*sqrt(7)*(4437922563568471220638688941619965174410065763500
*sqrt(7)*sqrt(5) - 26255445958792368545548556641042632173848895966799))/(4437922
563568471220638688941619965174410065763500*sqrt(7)*sqrt(5) - 2625544595879236854
5548556641042632173848895966799))*sqrt((32678*sqrt(7)*sqrt(5) - 361375)/(6748007
00*sqrt(7)*sqrt(5) - 4799048559)) + 64015*4805^(1/4)*sqrt(826)*sqrt(2*x + 1)*(51
625*sqrt(7) - 32678*sqrt(5))*sqrt((32678*sqrt(7)*sqrt(5) - 361375)/(674800700*sq
rt(7)*sqrt(5) - 4799048559)) - 1984465*1193983^(1/4)*(10*sqrt(7) - 97*sqrt(5))))
- 1193983^(1/4)*sqrt(31)*(51625*sqrt(7)*(5*x^2 + 3*x + 2) - 32678*sqrt(5)*(5*x^
2 + 3*x + 2))*log(7316*sqrt(5)*(1193983^(1/4)*4805^(1/4)*sqrt(826)*(285743283745
2652734362918220094532883072047512224503*sqrt(7)*sqrt(5) - 169046283204304255922
56747921672766019291595984758430)*sqrt(2*x + 1)*sqrt((32678*sqrt(7)*sqrt(5) - 36
1375)/(674800700*sqrt(7)*sqrt(5) - 4799048559)) + 14455*sqrt(5)*(443792256356847
1220638688941619965174410065763500*sqrt(7)*sqrt(5)*(2*x + 1) - 52510891917584737
091097113282085264347697791933598*x - 262554459587923685455485566410426321738488
95966799) + 14455*sqrt(7)*(4437922563568471220638688941619965174410065763500*sqr
t(7)*sqrt(5) - 26255445958792368545548556641042632173848895966799))/(44379225635
68471220638688941619965174410065763500*sqrt(7)*sqrt(5) - 26255445958792368545548
556641042632173848895966799)) + 1193983^(1/4)*sqrt(31)*(51625*sqrt(7)*(5*x^2 + 3
*x + 2) - 32678*sqrt(5)*(5*x^2 + 3*x + 2))*log(-7316*sqrt(5)*(1193983^(1/4)*4805
^(1/4)*sqrt(826)*(2857432837452652734362918220094532883072047512224503*sqrt(7)*s
qrt(5) - 16904628320430425592256747921672766019291595984758430)*sqrt(2*x + 1)*sq
rt((32678*sqrt(7)*sqrt(5) - 361375)/(674800700*sqrt(7)*sqrt(5) - 4799048559)) -
14455*sqrt(5)*(4437922563568471220638688941619965174410065763500*sqrt(7)*sqrt(5)
*(2*x + 1) - 52510891917584737091097113282085264347697791933598*x - 262554459587
92368545548556641042632173848895966799) - 14455*sqrt(7)*(44379225635684712206386
88941619965174410065763500*sqrt(7)*sqrt(5) - 26255445958792368545548556641042632
173848895966799))/(4437922563568471220638688941619965174410065763500*sqrt(7)*sqr
t(5) - 26255445958792368545548556641042632173848895966799)))/((51625*sqrt(7)*(5*
x^2 + 3*x + 2) - 32678*sqrt(5)*(5*x^2 + 3*x + 2))*sqrt((32678*sqrt(7)*sqrt(5) -
361375)/(674800700*sqrt(7)*sqrt(5) - 4799048559)))

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Sympy [A]  time = 161.306, size = 246, normalized size = 0.87 \[ - \frac{304 \left (2 x + 1\right )^{\frac{3}{2}}}{5 \left (- 992 x + 620 \left (2 x + 1\right )^{2} + 372\right )} - \frac{896 \left (2 x + 1\right )^{\frac{3}{2}}}{5 \left (- 6944 x + 4340 \left (2 x + 1\right )^{2} + 2604\right )} + \frac{608 \sqrt{2 x + 1}}{25 \left (- 992 x + 620 \left (2 x + 1\right )^{2} + 372\right )} - \frac{12096 \sqrt{2 x + 1}}{25 \left (- 6944 x + 4340 \left (2 x + 1\right )^{2} + 2604\right )} - \frac{304 \operatorname{RootSum}{\left (407144088666112 t^{4} + 3325152256 t^{2} + 11045, \left ( t \mapsto t \log{\left (\frac{33312534528 t^{3}}{235} + \frac{166784 t}{235} + \sqrt{2 x + 1} \right )} \right )\right )}}{25} - \frac{448 \operatorname{RootSum}{\left (19950060344639488 t^{4} + 498437272576 t^{2} + 10878125, \left ( t \mapsto t \log{\left (- \frac{11049511452672 t^{3}}{2205125} + \frac{307918256 t}{2205125} + \sqrt{2 x + 1} \right )} \right )\right )}}{25} + \frac{64 \operatorname{RootSum}{\left (1722112 t^{4} + 1984 t^{2} + 5, \left ( t \mapsto t \log{\left (- \frac{27776 t^{3}}{5} + \frac{108 t}{5} + \sqrt{2 x + 1} \right )} \right )\right )}}{25} + \frac{16 \operatorname{RootSum}{\left (1230080 t^{4} + 1984 t^{2} + 7, \left ( t \mapsto t \log{\left (9920 t^{3} + 8 t + \sqrt{2 x + 1} \right )} \right )\right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+2*x)**(5/2)/(5*x**2+3*x+2)**2,x)

[Out]

-304*(2*x + 1)**(3/2)/(5*(-992*x + 620*(2*x + 1)**2 + 372)) - 896*(2*x + 1)**(3/
2)/(5*(-6944*x + 4340*(2*x + 1)**2 + 2604)) + 608*sqrt(2*x + 1)/(25*(-992*x + 62
0*(2*x + 1)**2 + 372)) - 12096*sqrt(2*x + 1)/(25*(-6944*x + 4340*(2*x + 1)**2 +
2604)) - 304*RootSum(407144088666112*_t**4 + 3325152256*_t**2 + 11045, Lambda(_t
, _t*log(33312534528*_t**3/235 + 166784*_t/235 + sqrt(2*x + 1))))/25 - 448*RootS
um(19950060344639488*_t**4 + 498437272576*_t**2 + 10878125, Lambda(_t, _t*log(-1
1049511452672*_t**3/2205125 + 307918256*_t/2205125 + sqrt(2*x + 1))))/25 + 64*Ro
otSum(1722112*_t**4 + 1984*_t**2 + 5, Lambda(_t, _t*log(-27776*_t**3/5 + 108*_t/
5 + sqrt(2*x + 1))))/25 + 16*RootSum(1230080*_t**4 + 1984*_t**2 + 7, Lambda(_t,
_t*log(9920*_t**3 + 8*_t + sqrt(2*x + 1))))/5

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x + 1)^(5/2)/(5*x^2 + 3*x + 2)^2,x, algorithm="giac")

[Out]

integrate((2*x + 1)^(5/2)/(5*x^2 + 3*x + 2)^2, x)